Parallel-Shift Gamma Includes Cross Partial Derivatives

# Gamma of interest rate derivatives


# Gamma of interest rate derivatives












consider an interest rate derivative whose value $V$ depends on $n$ interest rates $r_1, \dots, r_n$. Hence $V$ is a function in $n$ variables $V(r_1, \dots, r_n)$. My question concerns the gamma $\gamma$ of this derivative with respect to parallel shifts.

Does it "always" hold that $$ \gamma(r_1, \dots, r_n) = \sum_{i = 1}^n \frac{\partial^2 V}{\partial r_i^2}(r_1, \dots, r_n). $$

To clarify the notation: I define the delta $\delta$ of the derivative with respect to parallel shifts as $$ \delta(r_1, \dots, r_n) = \lim_{h \rightarrow 0} \frac{V(r_1 + h, \dots, r_n + h) - V(r_1, \dots, r_n)}{h}, $$ and $\gamma$ is then defined as $$ \gamma(r_1, \dots, r_n) = \lim_{h \rightarrow 0} \frac{\delta(r_1 + h, \dots, r_n + h) - \delta(r_1, \dots, r_n)}{h}. $$

From a mathematical standpoint, I think the identity should not hold in general since mixed partial derivatives (i.e. $\frac{\partial^2 V}{\partial r_i r_j}$) should be considered as well. I also assume that $V$ is sufficiently smooth function such that all second partial derivatives exist and are continuous. However, literature seems to suggest that the identity holds in general. What do you guys think?

## Answer by user35980 (score 2)

https://quant.stackexchange.com/a/60724

It's my understanding that indeed the cross partial derivative terms do have a contribution - so you're correct to say that what u really have to work with is a gamma matrix $$\gamma=[\frac{d^2V}{dr_idr_j}]_{i,j}.$$ In essence, the cross partial terms $\frac{d^2}{dr_idr_j}$ allude to the correlation between the various rates $(r_1,...,r_n)$. Assuming a high (c. 90%) correlation these terms generally can be ignored. For this reason the gamma as computed by your stated formula is a kind of 'local' gamma as opposed to a 'global' gamma given by the entire matrix. For calculating this 'global' gamma, banks generally use some form of principal components analysis (or some other linear transformation) as it's a less computationally expensive way of capturing the majority of the variance in rate sensitivity without having to compute the whole matrix.

## Answer by John (score 0)

https://quant.stackexchange.com/a/68666

Correct. You're starting with specifying the scenario where all rates move parallel. The correlation implicit there is 100%. There is nothing stochastic left. The gamma to a parallel shift is as cited originally. No matrix is needed.